Hi Bass. Great question. I will defer others with more expertise on figs for a specific answer, but here is a general answer: Genes are constantly in flux. Even a cloned plant might experience a genetic mutation over time caused by a virus, etc., so that even with asexual reproduction there will be small genetic shifts over time.
Further, as Jon has said, figs are very sensitive to their environment, so even if you had two figs that were genetically identical (at the time), they might appear somewhat different due to variations in local environmental factors, and the fruit might be dramatically different.
It is good to have names, so we have some reasonable agreement about what it is we are talking about, and people know what they are getting and what they are growing.
But while the abstraction of perfect categories is appealling, it is likely the the reality is far too complex to fit into neat and separate categories on every plant in perpetuity. There may have been, for example, an original Mission Fig, but now there are many, including some that were mislabled and are not related to the original. But they are all Mission Fig-ish.
And Jason, we have that same pattern with all of our inground trees: Freeze back and regrowth from the roots every year. But after a few years, they toughen up and form a solid hard wood base that take the freezing winters.
Best wishes to all.
John